Homework5

1. (Problem 4.37 from Proakis 5th Edition)
This problem deals with the characteristics of a DPSK signal.

  • Suppose we wish to transmit the data sequence 1 1 0 1 0 0 0 1 0 1 1 0 by binary DPSK. Let $s(t)= A cos(2 \pi f_c t + \theta)$ represent the transmitted signal in any signaling interval of duration $T$. Give the phase of the transmitted signal for the data sequence. Begin with $\theta= 0$ for the phase of the first bit to be transmitted.
  • If the data sequence is uncorrelated, determine and sketch the power density spectrum of the signal transmitted by DPSK.

2. (Problem 3.19 from Proakis 5th Edition)
The information sequence $\{a_n\}^{\infty}_{n=-\infty}$ is a sequence of iid random variables, each taking values $+1$ and $-1$ with equal probability. This sequence is to be transmitted at baseband by a biphase coding scheme, described by

(1)
\begin{align} s(t) = \sum^{\infty}_{n=-\infty} a_n g(t-nT) \end{align}

where g(t) is shown in Figure P3.19.

Figure%20P3.19.png
  • Find the power spectral density of $s(t)$.
  • Assume that it is desirable to have a zero in the power spectrum at $f=1/T$. To this end, we use a precoding scheme by introducing $b_n=a_n + k a_{n-1}$, where $k$ is some constant, and then transmit the $\{b_n\}$ sequence using the same $g(t)$. Is it possible to choose $k$ to produce a frequency null at $f=1/T$? If yes, what are the appropriate values and the resulting power spectrum?
  • Now assume we want to have zeros at all multiples of $f_0=1/4T$. Is it possible to have these zeros with an appropriate choice of $k$ in the previous part? If not, then what kind of precoding do you suggest to achieve the desired result?

3. (Problem 3.22 from Proakis 5th Edition)
A digital signaling scheme is defined as

(2)
\begin{align} X(t)= \sum^{\infty}_{n=-\infty}[a_n u(t-nT) cos(2 \pi f_c t)- b_n u(t-nT) sin(2 \pi f_c t)] \end{align}

where $u(t)= \Lambda(t/2T)$,

(3)
\begin{align} \Lambda(t)= \left\{ \begin{array}{ll} t+1 & -1 \leq t \leq 0 \\ -t+1 & 0 \leq t \leq 1 \\ 0 & \text{otherwise} \end{array} \right. \end{align}

and each $(a_n, b_n)$ pair is independent from the others and is equally likely to take any of the three values $(0, 1)$, $(\sqrt{3}/2, -1/2)$, and $(-\sqrt{3}/2, -1/2)$.

  • Determine the lowpass equivalent of the modulated signal. Determine the in-phase and quadrature components.
  • Determine the power spectral density of the lowpass equivalent signal; form this determine the power spectral density of the modulated signal.
  • By employing a precoding scheme of the form
(4)
\begin{align} \left\{ \begin{array}{l} c_n= a_n+ \alpha a_{n-1} \\ d_n= b_n+ \alpha b_{n-1} \\ \end{array} \right. \end{align}

where $\alpha$ is in general a complex number, and transmitting the signal

(5)
\begin{align} Y(t)= \sum^{\infty}_{n=-\infty}[c_n u(t-nT) cos(2 \pi f_c t)- d_n u(t-nT) sin(2 \pi f_c t)] \end{align}

we want to have a lowpass signal that has no dc component. Is it possible to achieve this goal by an appropriate choice of $\alpha$? If yes, find this value.

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